Learning Objectives
- Understand order and degree of differential equations
- Learn to form differential equations from general solutions
- Solve DEs using variable separable, homogeneous, and linear methods
Key Concepts
Order and Degree
Order: Highest order derivative in the equation. Degree: Highest power of the highest order derivative (when equation is polynomial in derivatives). If equation involves sin(dy/dx), edy/dx, etc., degree is not defined. Number of arbitrary constants in general solution = Order of DE.
Formation of Differential Equations
Given a family of curves with n arbitrary constants, differentiate n times and eliminate all constants to get the DE. Example: y = aex + be-x (2 constants → order 2 DE). Differentiate twice and eliminate a, b to get y'' - y = 0.
Methods of Solving
1. Variable Separable: Form: f(x)dx = g(y)dy. Integrate both sides separately. Example: dy/dx = x/y → y dy = x dx → y2/2 = x2/2 + C.
2. Homogeneous Differential Equations: Form: dy/dx = f(y/x). Substitute y = vx, then dy/dx = v + x(dv/dx). Equation reduces to variable separable in v and x. A function f(x,y) is homogeneous of degree n if f(tx, ty) = tnf(x,y).
3. Linear Differential Equations: Form: dy/dx + Py = Q where P and Q are functions of x only. Integrating Factor (IF) = e∫P dx. Solution: y × IF = ∫(Q × IF)dx + C. Similarly for dx/dy + Px = Q (P, Q functions of y): IF = e∫P dy, solution: x × IF = ∫(Q × IF)dy + C.
Important Examples
Newton's law of cooling: dT/dt = -k(T - Ts). Population growth: dP/dt = rP. Radioactive decay: dN/dt = -λN. All are first-order linear or separable DEs.
Summary
Differential equations relate functions to their derivatives. The order equals the number of arbitrary constants in the general solution. Key solution methods are variable separable, homogeneous substitution (y = vx), and linear equations using integrating factors.
Important Terms
- Order: Highest order derivative present
- Degree: Power of highest order derivative (when polynomial)
- General solution: Contains arbitrary constants equal to order
- Particular solution: General solution with specific constant values
- Integrating Factor: e∫Pdx for linear DE dy/dx + Py = Q
- Homogeneous DE: Can be written as dy/dx = f(y/x)
Quick Revision
- Order = highest derivative; Degree = power of highest derivative (if polynomial)
- n arbitrary constants → order n DE
- Variable separable: separate x and y terms, integrate
- Homogeneous: substitute y = vx, then separate v and x
- Linear: dy/dx + Py = Q; IF = e∫Pdx; y·IF = ∫Q·IF dx + C
- Check: degree undefined if derivatives appear in sin, exp, log, etc.